Final answer:
To find the center of mass of the region, we need to calculate the area bounded by the graph of f(x) = 1/(x-2) and below the x-axis and find the x-coordinate of the centroid. The center of mass can be calculated by dividing the total moment by the total mass.
Step-by-step explanation:
In order to find the center of mass of the region bounded by the graph of f(x) = 1/(x-2) and below the x-axis over the interval [3,8], we need to calculate the area bounded by this graph and find the x-coordinate of the centroid. Since the graph of f(x) extends to infinity as x approaches 2, we can split the region into two parts: one above the x-axis and one below the x-axis.
Using the formula for the area between the graph and the x-axis, we can calculate the area as follows:
Above the x-axis:
− ∫ (1/(x-2)) dx with limits of integration from 3 to 8
Below the x-axis:
∫ (1/(x-2)) dx with limits of integration from 3 to 8
To find the center of mass, we divide the total moment by the total mass. The moment is the integral of x times the density function, and the mass is the integral of the density function. Therefore, the x-coordinate of the center of mass can be calculated as follows:
x-coordinate of the center of mass = (∫ (x * (1/(x-2))) dx from 3 to 8) / (− ∫ (1/(x-2)) dx from 3 to 8)